Theorem ltadd2dd

Description: Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	letrd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	ltletrd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	ltadd2dd	⊢ ( 𝜑 → ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		letrd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		ltletrd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
5	1 2 3	ltadd2d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) ) )
6	4 5	mpbid	⊢ ( 𝜑 → ( 𝐶 + 𝐴 ) < ( 𝐶 + 𝐵 ) )